Torsions and integrations

ABSTRACT

The solutions of a number of well-known boundary value problems of complex analysis (for example, the Riemann boundary value problem) can be found in the form of curvilinear integrals over the boundaries of domains under consideration. In this connection, the classical results on that problems concern domains with rectifiable boundaries only. On the other hand, the boundary value problems themselves make sense for non-rectifiable boundaries, too. This is the reason for recent development of theory of generalized integration over plane non-rectifiable Jordan curves and arcs. The existence of such generalized integrations over non-rectifiable arcs depends on certain geometry properties of the arcs in neighborhoods of their ends. Here we consider the connections of generalized integration and so-called torsions of the path of integration at its end points.

Keywords:

• Riemann boundary value problem
• jump problem
• generalized integration
• non-rectifiable arc
• torsion
• Cauchy-type integral

AMS SUBJECT CLASSIFICATION:

• 30E25

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported as part of the development program of the Scientific and Educational Mathematical Center of the Volga Federal District, agreement No. 075-02-2020-1478.